If you discard a card with madness to pay the cost of a spell or activated ability, that card's madness triggered ability (and the spell that card becomes, if you choose to cast it) will resolve before the spell or ability the discard paid for.
Asylum Visitor
Cards are discarded in a Magic game only from a player's hand. Effects that put cards into a player's graveyard from anywhere else do not cause those cards to be discarded.
If you choose not to cast a card with madness when the madness triggered ability resolves, it's put into your graveyard. Madness doesn't give you another chance to cast it later.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Bastien L. Deharme
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
